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* Current affiliation: National Center for Atmospheric Research

Explicit Treatment of Model Error Simultaneous State and Parameter Estimation with an Ensemble Kalman Filter Altuğ Aksoy*, Fuqing Zhang, and John W. Nielsen-Gammon Texas A&M University. * Current affiliation: National Center for Atmospheric Research.

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* Current affiliation: National Center for Atmospheric Research

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  1. Explicit Treatment of Model ErrorSimultaneous State and Parameter Estimation with an Ensemble Kalman FilterAltuğ Aksoy*, Fuqing Zhang, and John W. Nielsen-GammonTexas A&M University * Current affiliation: National Center for Atmospheric Research

  2. The Sea Breeze Model: Equations *(Aksoy et al. 2005, JGR) • Two-dimensional, irrotational, incompressible flow with prognostic variables buoyancy (b′, perturbation tempertaure) and vorticity (η′): • Explicit heating function: • Estimated model parameters: Heating amplitude Heating depth Mean horizontal wind Vertical diffusion coefficients Static stability * Similar to Rotunno’s (1983) linear approach

  3. The Sea Breeze Model: Numerics • Model domain: • Numerical features: • Leapfrog time integration • Cranck-Nicholson implicit trapezoidal vertical diffusion • Rayleigh-damping sponge layers for vorticity • Second-order lagged horizontal diffusion for both model variables • Asselin-type filtering to control computational mode of the leapfrog scheme Sponge Layer 2 km 300 km 500 km 300 km Grid resolution: Horizontal: 4 km Vertical: 50 m Sponge Layer Sponge Layer Forecast Domain 3 km Land Sea

  4. 3 km 3 km Surface Surface 250 km 250 km 500 km 500 km Sea Sea Land Land 3 km Sea Land Surface 250 km 500 km The Sea Breeze Model: Perfect-model behavior Temperature Vorticity Winds and Streamfunction 48H Forecast Noon Maximum Heating

  5. 3 km 3 km Surface Surface 250 km 250 km 500 km 500 km Sea Sea Land Land 3 km Sea Land Surface 250 km 500 km The Sea Breeze Model: Perfect-model behavior Temperature Vorticity Winds and Streamfunction 51H Forecast 3:00PM Onset of Sea Breeze

  6. 3 km 3 km Surface Surface 250 km 250 km 500 km 500 km Sea Sea Land Land 3 km Sea Land Surface 250 km 500 km The Sea Breeze Model: Perfect-model behavior Temperature Vorticity Winds and Streamfunction 54H Forecast 6:00PM Warmest Temperature

  7. 3 km 3 km Surface Surface 250 km 250 km 500 km 500 km Sea Sea Land Land 3 km Sea Land Surface 250 km 500 km The Sea Breeze Model: Perfect-model behavior Temperature Vorticity Winds and Streamfunction 57H Forecast 9:00PM Strongest Sea Breeze

  8. 3 km 3 km Surface Surface 250 km 250 km 500 km 500 km Sea Sea Land Land 3 km Sea Land Surface 250 km 500 km The Sea Breeze Model: Perfect-model behavior Temperature Vorticity Winds and Streamfunction 60H Forecast Midnight Maximum Cooling

  9. 3 km 3 km Surface Surface 250 km 250 km 500 km 500 km Sea Sea Land Land 3 km Sea Land Surface 250 km 500 km The Sea Breeze Model: Perfect-model behavior Temperature Vorticity Winds and Streamfunction 63H Forecast 3:00AM Onset of Land Breeze

  10. 3 km 3 km Surface Surface 250 km 250 km 500 km 500 km Sea Sea Land Land 3 km Sea Land Surface 250 km 500 km The Sea Breeze Model: Perfect-model behavior Temperature Vorticity Winds and Streamfunction 66H Forecast 6:00AM Coldest Temperature

  11. 3 km 3 km Surface Surface 250 km 250 km 500 km 500 km Sea Sea Land Land 3 km Sea Land Surface 250 km 500 km The Sea Breeze Model: Perfect-model behavior Temperature Vorticity Winds and Streamfunction 69H Forecast 9:00AM Strongest Land Breeze

  12. 3 km 3 km Surface Surface 250 km 250 km 500 km 500 km Sea Sea Land Land 3 km Sea Land Surface 250 km 500 km The Sea Breeze Model: Perfect-model behavior Temperature Vorticity Winds and Streamfunction 72H Forecast Noon Maximum Heating

  13. Model Error - Enkf Properties(Aksoy et al. 2006, MWR) • Observations: Surface buoyancy observations on land • Observational error: Standard deviation of 10-3 ms-2 • Observation spacing: 40 km (10 grid points) • Ensemble size: 50 members • Ensemble initialization: Perturbations from model climatology • Covariance localization: Gaspari and Cohn’s (1999) fifth-order correlation function with 100 grid-point radius of influence • Observation processing: Sequential with no correlation between observation errors (Snyder and Zhang 2003) • Filter: Square-root after Whitaker and Hamill (2002) with no perturbed observations

  14. Perfect-Model EnKF Results 3H Prior 3H Posterior Buoyancy 3H Prior 3H Posterior Vorticity

  15. Perfect-Model EnKF Results 36H Prior 36H Posterior Buoyancy 36H Prior 36H Posterior Vorticity

  16. Estimation Performance Buoyancy Vorticity MRE = 60% MRE = 54%

  17. Mean horizontal wind Static stability Vorticity diff. coef. Buoyancy diff. coef. Heating amplitude Heating depth Estimation Performance

  18. Parameter Identifiability Mean horizontal wind Static Stability : RMS correlation M : Any spatial domain  :Any parameter b : Buoyancy • Distinct differences among parameters • Static stability and heating depth sensitive to observation location • Vort. diff. coef. with smallest correlation, appears to exhibit smallest identifiability Vorticity Diffusion Coef. Heating Depth

  19. MM5 Experiments: Experimental Setup(Aksoy et al. 2006, GRL, submitted) • 36-km resolution with55×55 grid-point domain • 43 vertical sigma layers with50 hPa model top • Initialized: 00Z 28 Aug 2000 • Parameterizations: • MRF PBL scheme • Grell cumulus scheme withshallow cumulus option • Simple-ice microphysicalparameterization • Prognostic variables • Winds (u, v, w ) • Temperature (T ) • Water vapor mixing ratio (q ) • Pressure perturbation (p ‘ )

  20. Control Forecast • Evidence of the clockwise turning of winds • Penetration of the temperature and moisture gradients inland during the sea breeze phase • Well-established return flow during the land breeze phase Sea breeze phase (7pm local) Land breeze phase

  21. Ensemble and Filter Properties • Ensemble size: 40 members • Observations: Surface and sounding observations of u, v, and T • Observational error: Std. deviations of 2 ms-1 for u and v ; 1 K for T • Observation spacing: 72 km for surface, 324 km for sounding • Covariance localization: Gaspari and Cohn’s (1999) fifth-order correlation function with 30 grid-point radius of influence • Observation processing: Sequential (Snyder and Zhang 2003) • Filter: Square-root (Whitaker and Hamill 2002)

  22. Parameter Estimation Details • Not attempting to identify individual error sources within the PBL scheme associated with different empirical parameters: • Multiplier (m) of the vertical eddy mixing coefficient implanted into the MRF PBL code → for a value of 1.0, the original MRF PBL computation is simply repeated • Variance limit applied at 1/4 of initial parameter error • Updating is carried out spatially: • Prior parameter value converted to 2-d matrix assumed at surface • Spatial updating is performed with same covariance localization properties as the updating of the state • Updated 2-d distribution is averaged to obtain posterior global parameter value

  23. Correlation Signal – (T,m) and (U,m) • Relatively strong overall correlation signal with both temperature and winds (signal strength “comparable” to idealized sea breeze model experiments) • Spatially and temporally varying correlation structure • Stronger signal near the surface • Smaller-scale variability with horizontal winds

  24. Estimation Performance – 3 Cases Initial Mean Error = +0.2 Initial Mean Error = +0.65 Initial Mean Error = -0.3

  25. Concluding Remarks • EnKF demonstrated to be promising for explicit treatment of model error through simultaneous state and parameter estimation • Lessons from the idealized sea breeze model experiments: • Sensitivity to observation location, radius of influence, and variance limit is parameter-specific • Counter-acting correlations do lead to identifiability issues with some parameter pairs (do we really need to estimate every single parameter?) • A more global approach to the MRF PBL scheme in MM5 appears to be responding well • Updating of a global parameter through observations that contain spatial information is an issue and does lead to divergence as number of observations increases: • We have approached this problem through our “spatial updating” technique – ad hoc but effective

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